On the existence of fixed points and ergodic retractions for Lipschitzian semigroups in Hilbert spaces

Abstract

and especially nonexpansive in the case of k = 1. Let S be a semitopological semigroup and let S = (c: s E S) be a continuous representation of S as Lipschitzian mappings of C into itself, which is said to be a Lipschitzian semigroup on C. A Lipschitzian semigroup S = (T,: s E S] with Lipschitz constants k,, s E S is called a uniformly k-Lipschitzian semigroup and a nonexpansive semigroup when k, = k and k, = 1 for all s E S, respectively. Goebel and Kirk [7] first called a mapping T of C into itself a uniformly k-Lipschitzian mapping on C in the case when S = (T”: n E N] is a Lipschitzian semigroup with k, = k for all n E N. Lifschitz [12] obtained a fixed point theorem for uniformly k-Lipschitzian mappings on C with k < ~5, and this result was extended to uniformly k-Lipschitzian semigroups which are left reversible by Downing and Ray [6]. Lau [l l] showed that a nonexpansive semigroup has a common fixed point if there exists a left invariant mean on the space RUG’(S) of bounded right uniformly continuous functions on the semigroup. Furthermore, Ishihara and Takahashi [9] extended Lau’s result to uniformly k-Lipschitzian semigroups and also gave a simpler proof of Downing and Ray’s theorem. On the other hand, the first nonlinear ergodic theorem for nonexpansive mappings was established by Baillon [ 11: Let T be a nonexpansive mapping of C into itself and suppose that the set F(T) of fixed points of T is nonempty. The the Cesaro means